Changing magnetic field causing current.

I have read about how magnetic forces are a consequence of special relativity when you have something moving in a magnetic field, like how moving a wire in a magnetic field causes a current and it can be summarized by using magnetic flux and so on (magnetic flux has area in it, which takes into account length of the wire and the speed of it as if the area is changing etc). Then it is also said that any component of the magnetic flux can change, even the magnetic field, and the effect is still the same.

But they seem different. Is there an explanation as to why a changing magnetic field causes the same effect as moving the wire, using relativity or other theories?

Are you saying that moving a loop of wire into or out of a magnetic field to change the flux (and induce a voltage) seems different than changing the magnitude of the field through the loop there by also creating a voltage? The only thing that Faraday's law says is the time rate of change of the amount of magnetic flux through a loop (real or imaginary) creates a (negative) voltage. So as you said the formula for magnetic flux is Φ=BA. and Faraday says dΦ/dt makes an electric field. This leaves you free to change either one. If you choose to move along with the loop (be in its frame of reference) then the field has moved (i.e. changed magnitude). If you stay with the field (i.e. its source is stationary to you) then the loop has moved. Either way Faraday's law is valid and a voltage will be induced. Relativity just says either one is free to say the other is moving, therefore the law must work both ways, or there'd be a preferred frame of reference.

If I have two parallel wires with current going the same way, then there is the argument that if you consider the electron's rest frame the protons in the other wire are length contracted, therefore more densely packed than the other wire's electrons and because of that the electrons in the first wire feel an electric force from it.

But if electrons are still in one wire and accelerating in the other wire (ie the strength of the magnetic field changes and so also the flux), what argument is used to explain that? It seems to me that the same explanation wouldn't work, because the electrons in one wire are still, but perhaps there are some other relativistic considerations or explanations?

I am familiar with the first part you said. If there are two currents parallel the wires are attracted to one another. In the rest frame of the wires its because of the magnetic fields coming together, in the electrons frame its a static electric force due to charge densities b/c of contraction. (I realize you know this already...) but in the case where the second wire (let's call it a loop) has no current there will be no attraction. The electrons in that wire will see a magnetic field but in the lab frame they aren't moving and so no Lorentz force. The loop will see the magnetic field though and it the current changes it will see the change in field strength and have a nonconservative electric field induced around the loop. The static field they had in the first scenario and the one induced in the loop are very different. The former was toward the other wire and conservative. The latter is around the loop (closed loop integral), nonconservative, and induces a current.

Why is the electric field induced? Is there a relativistic explanation for it? I ask for a relativistic explanation because for the moving charges case it shows nicely how the force is just a consequence of relativity. Is there a similar way to show that the induced electric field is a consequence of relativity, or some other explanation why a changing magnetic field causes an electric field?

My understanding of that is only so-so. The derivation I've seen involved the Lorenz force, divided by q, being broken into two parts:
F/q = E + v/q x B.
The velocity refers to a differential segment of the loop and B is the local field there. These two parts on the right side are the the part due to changing fields (E), and the part due to to 'moving' the loop (v/q x B). You must integrate each one, and that's where my recollection breaks down. It involves vector potentials and I'm a little new to these. The last thing I want to do is try to tell you about something I'm not qualified to explain.

"Is there an explanation as to why a changing magnetic field causes the same effect as moving the wire."
it's because they are equivalent effects which results from different frames of reference.

The electric field E at any point, and the magnetic field B, aren't by themselves tensors. But when combined properly, they do form a tensor, the Faraday tensor.

Because E and B aren't themselves tensors, different observers will ascribe proper acceleration as being due to solely electric fields, or a combination of electric and magnetic fields. That the Faraday tensor is actually a tensor and therefore transforms as a tensor describes exactly how the electromagnetic field transforms.

That Wikipedia article is interesting, but it seems to talk about the cases when the current and charge both are moving, so I wondered if there is a similar explanation for when the charge doesn't move but is in a changing magnetic field.

Like for example, if I have current through a wire and an electron moving parallel to it, it feels a force that can be explained by relativity. If the electron is at rest relative to the wire, while current goes through the wire, it feels no force and that is also explained by relativity, i.e. no movement, therefore no length contraction or other considerations, and since the wire is electrically neutral in its own frame there is no force. But, if the electron is still compared to the wire and the current through the wire is accelerating, the electron feels a force. I am interested in finding out about the relativistic or other considerations that result in such an effect when the current is accelerating.

That Wikipedia article is interesting, but it seems to talk about the cases when the current and charge both are moving, so I wondered if there is a similar explanation for when the charge doesn't move but is in a changing magnetic field.

Like for example, if I have current through a wire and an electron moving parallel to it, it feels a force that can be explained by relativity. If the electron is at rest relative to the wire, while current goes through the wire, it feels no force and that is also explained by relativity, i.e. no movement, therefore no length contraction or other considerations, and since the wire is electrically neutral in its own frame there is no force. But, if the electron is still compared to the wire and the current through the wire is accelerating, the electron feels a force. I am interested in finding out about the relativistic or other considerations that result in such an effect when the current is accelerating.

You seem to be under the impression that the magnetic force isn't "real." If you start in some frame where the magnetic field is larger than the electric field (i.e. Bc>E), then you will never be able to transform to a reference frame where the magnetic field completely vanishes. So it's not possible to completely explain the magnetic force in terms of the electric force + relativity, though it does work for some specific situations.